Abstract

This paper investigates the increased stability behavior commonly observed in low-speed machining. In the past, this improved stability has been attributed to the energy dissipated by the interference between the workpiece and the tool relief face. In this study, an alternative physical explanation is described. In contrast to the conventional approach, which uses a point force acting at the tool tip, the cutting forces are distributed over the tool-chip interface. This approximation results in a second-order delayed integrodifferential equation for the system that involves a short and a discrete delay. A method for determining the stability of the system for an exponential shape function is described, and temporal finite element analysis is used to chart the stability regions. Comparisons are then made between the stability charts of the point force and the distributed force models for continuous and interrupted turning.

Tlusty, J., and Polacek, M., 1963, “The Stability of Machine Tools Against Self-Excited Vibrations in Machining,” "Proceedings of the ASME International Research in Production Engineering", pp. 465–474.

Figures

Schematic diagram of the interrupted cutting process. The force is proportional to the uncut chip area when the tool is in contact with the work piece but drops to zero when the tool vibrates freely (31).

Short time delay embedding into the force model. The area of a certain segment of the chip flowing over the rake face in (b) can be described by the area of the same segment as it was being cut at the tool tip in (a) at time t−t̂.

Stability charts for the point force model of continuous turning plotted as a function of the nondimensionalized cutting speed and depth of cut. The damping ratios used are (a) ζ=0.0038 and (b) ζ=0.02 (unstable regions shaded).

Stability charts for the point force model of interrupted turning plotted as a function of the nondimensionalized cutting speed and depth of cut. The damping ratio used is ζ=0.0038 and the cases for different fractions of the workpiece revolution that the tool is cutting are (a) ρ=0.05, (b) ρ=0.10, and (c) ρ=0.20 (unstable regions shaded).

Schematic diagram of a: (a) distributed force model and (b) conventional point force model. Case (a) uses a stress distribution over the tool rake face and applies a finite time for the chip to travel along the tool-chip interface. Case (b) is the conventional modeling approach of using a point force and a discrete delay model.

Stability charts for the distributed force model of continuous turning plotted as a function of the nondimensionalized cutting speed and depth of cut. The delay ratio used is r=0.001 and the damping ratios used are (a) ζ=0.0038 and (b) ζ=0.02 (unstable regions shaded).

Stability charts for the distributed force model of continuous turning plotted as a function of the nondimensionalized cutting speed and depth of cut. The damping ratio used is ζ=0.0038 and the delay ratios used are (a) r=0.03, (b) r=0.05, and (c) r=0.10 (unstable regions shaded).

Stability charts for the distributed force model of interrupted turning plotted as a function of the nondimensionalized cutting speed and depth of cut. The damping ratio used is ζ=0.0038 and the cases for different fractions of the workpiece revolution that the tool is cutting and different delay ratios are (a) ρ=0.05 and r=0.001, (b) ρ=0.10 and r=0.001, (c) ρ=0.20 and r=0.001, (d) ρ=0.05 and r=0.03, (e) ρ=0.10 and r=0.03, and (f) ρ=0.20 and r=0.03 (unstable regions shaded).

Stability charts for the distributed force model of interrupted turning plotted as a function of the nondimensionalized cutting speed and depth of cut. The damping ratio used is ζ=0.0038 and the cases for different fractions of the workpiece revolution that the tool is cutting and different delay ratios are (a) ρ=0.05 and r=0.05, (b) ρ=0.10 and r=0.05, (c) ρ=0.20 and r=0.05, (d) ρ=0.05 and r=0.10, (e) ρ=0.10 and r=0.10, and (f) ρ=0.20 and r=0.10 (unstable regions shaded).

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